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Properties

Label 2-369-41.21-c1-0-6
Degree $2$
Conductor $369$
Sign $0.0985 - 0.995i$
Analytic cond. $2.94647$
Root an. cond. $1.71653$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.05 + 0.666i)2-s + (2.14 + 1.55i)4-s + (−1.72 + 2.37i)5-s + (−1.11 + 2.18i)7-s + (0.828 + 1.13i)8-s + (−5.12 + 3.72i)10-s + (6.07 + 0.961i)11-s + (0.531 − 0.270i)13-s + (−3.74 + 3.74i)14-s + (−0.700 − 2.15i)16-s + (−0.0921 + 0.581i)17-s + (−1.67 − 0.854i)19-s + (−7.40 + 2.40i)20-s + (11.8 + 6.01i)22-s + (1.21 − 3.75i)23-s + ⋯
L(s)  = 1  + (1.45 + 0.471i)2-s + (1.07 + 0.779i)4-s + (−0.771 + 1.06i)5-s + (−0.420 + 0.825i)7-s + (0.292 + 0.402i)8-s + (−1.61 + 1.17i)10-s + (1.83 + 0.289i)11-s + (0.147 − 0.0750i)13-s + (−0.999 + 0.999i)14-s + (−0.175 − 0.539i)16-s + (−0.0223 + 0.141i)17-s + (−0.384 − 0.196i)19-s + (−1.65 + 0.538i)20-s + (2.51 + 1.28i)22-s + (0.254 − 0.782i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0985 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0985 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.0985 - 0.995i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ 0.0985 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87579 + 1.69925i\)
\(L(\frac12)\) \(\approx\) \(1.87579 + 1.69925i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + (-1.29 - 6.27i)T \)
good2 \( 1 + (-2.05 - 0.666i)T + (1.61 + 1.17i)T^{2} \)
5 \( 1 + (1.72 - 2.37i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.11 - 2.18i)T + (-4.11 - 5.66i)T^{2} \)
11 \( 1 + (-6.07 - 0.961i)T + (10.4 + 3.39i)T^{2} \)
13 \( 1 + (-0.531 + 0.270i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.0921 - 0.581i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (1.67 + 0.854i)T + (11.1 + 15.3i)T^{2} \)
23 \( 1 + (-1.21 + 3.75i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.785 + 4.96i)T + (-27.5 + 8.96i)T^{2} \)
31 \( 1 + (-4.03 + 2.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.87 + 1.36i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-2.50 - 0.812i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.33 + 4.57i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (0.818 + 5.16i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (3.23 - 9.94i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.968 + 0.314i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.42 - 0.542i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (4.74 + 0.751i)T + (67.5 + 21.9i)T^{2} \)
73 \( 1 - 0.596iT - 73T^{2} \)
79 \( 1 + (-3.13 - 3.13i)T + 79iT^{2} \)
83 \( 1 + 3.79T + 83T^{2} \)
89 \( 1 + (-5.50 + 10.8i)T + (-52.3 - 72.0i)T^{2} \)
97 \( 1 + (10.0 - 1.58i)T + (92.2 - 29.9i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.83417179572474191574836550538, −11.21384985654630482519993453368, −9.813150871228946283007222775732, −8.744510287873785120063347198992, −7.37722338778203120309001441596, −6.51007193054242937413852785301, −6.06126019606435275498776554416, −4.48827370973808164140543688211, −3.70858117385574082631222144729, −2.67224839402122445840252699928, 1.30540845638355610012891214108, 3.45029253853048012766648946375, 4.04431922887653534541764265466, 4.87995796982188058035388979958, 6.14947233105814099318502128543, 7.09452734606683344396038067768, 8.507702348183248083158130678079, 9.307487363063399654228307468099, 10.70911083909002023136095901480, 11.58302193805862704671736185596

Graph of the $Z$-function along the critical line